A recent polynomial version of the celebrated Sarnak’s conjecture asked whether, given a nonlinear polynomial $p \in \mathbb{Z}[x]$, zero entropy minimal topological dynamical system $(X,T)$, $f \in C(X)$, and $x_0 \in X$, the sequence $f(T^{p(x)} x_0)$ is uncorrelated with the Mobius function $\mu$.

This conjecture is false, and has been refuted in two recent works with interesting and somewhat difficult constructions. However, we can use a simple symbolic construction to prove the following: when $(k_n)$ has zero Banach density, then not only may the sequence $f(T^{k_n} x_0)$ be correlated with $\mu$, there are actually no restrictions on the sequence whatsoever.